Chaos: Making a New Science - by James Gleick
My Notes
The Butterfly Effect
Chaos emerges as a new approach to science in the latter half of the 20th century.
“Yes, you could change the weather. You could make it do something different from what it would otherwise have done. But if you did, then you would never know what it would otherwise have done. It would be like giving an extra shuffle to an already well-shuffled pack of cards. You know it will change your luck, but you don’t know whether for better or worse.”
Lorenz’s computer simulation of the weather showed that it’s impossible to predict the weather more than a few days ahead because of The Butterfly Effect. The Butterfly Effect happens when a tiny input change results in large output change. The weather is an example of non-linear behavior.
It my be counterintuitive, but small changes in input don’t necessarily result in small changes in output.
Computer programs exhibit the Butterfly Effect. For example, when floating point calculations round off a few tiny decimals, that small change can propagate through the system and cause wildly different output.
The Butterfly Effect shows up when heating fluids. Apply some heat and the fluid starts rolling in cycles. Add more heat, and the rolls start to wobble. At very high temperatures, “the flow becomes wild and turbulent.” Again, this is non-linearity at work.
Graphs of this chaotic Butterfly Effect behavior may show beautiful, complex patterns. There is order within chaos. Google “Lorenz Attractor” and see for yourself.
Revolution
Chaos Theory was shunned by most 20th century scholars. Chaos threatens all that they have built there careers upon.
Science has traditionally ignored small nonlinearities (i.e. friction, air resistance), to keep the equations and calculations simple and understandable. The Butterfly Effect shows that neglecting small nonlinearities can cause the science to be drastically wrong.
Simple oscillators such as swings and pendulums can exhibit wild, unpredictable behavior.
What does “stable” mean? Can chaotic systems exhibit some sense of stability, if viewed in a useful way?
Smale studies dynamic systems, topology and dynamic systems. By analyzing oscillators in their phase space, he sees beautiful patterns.
The mysterious swirling Great Red Spot of Jupiter is explained through Chaos Theory. Marcus programs a computer simulation of the spot in “small-scale chaotic flow.”
“Marcus learned that a deterministic system can produce much more than just periodic behavior. He knew to look for wild disorder, and he knew that islands of structure could appear within the disorder. So he brought the problem of the Great Red Spot an understanding that a complex system can give rise to turbulence and coherence at the same time. He could work within an emerging discipline that was creating its own tradition of using the computer as an experimental tool.”
Life’s Ups and Downs
Scientists learn not to see chaos, because it disagrees with their need to predict and understand things.
“A man leaves the house in the morning thirty seconds late, a flowerpot misses his head by a few millimeters, and then he is run over by a truck. Or less dramatically, he misses a bus that runs every ten minutes - his connection to a train that runs every hour. Small perturbations in one’s daily trajectory can have large consequences… Science, though - science was different.”
Chaotic patterns emerge in population growth and decline.
Some populations rise and fall less and less over time, until they reach an equilibrium with their environment.
Other systems grow until they become chaotic and then split in two. The resulting bifurcations grow until they become chaotic and split again. We see the periods keep doubling - 1, 2, 4, 8, 16, 32.
Population biology and ecology exhibit similar periods. Bizarre behavior like this exists throughout nature, which is full of nonlinear systems.
A Geometry of Nature
An eccentric guy named Mandelbrot dabbles in many fields - economics, mathematics, game theory, etc - and brings a unique approach to each.
Working at IBM, Mandelbrot analyzes the noise in telephone lines. What is the pattern? He sees a Cantor Set, a pattern that repeats itself throughout different scales (but may confuse us when we are stuck in one view scale).
Mandelbrot discovers irregularities in the distance of country’s borders. He shows that “any coastline is - in a sense - infinitely long. In another sense, the answer depends on the length of your ruler.” As you go to finer and finer scales, there is more and more distance to measure. Think about this for a few minutes. It’s mind-blowing.
Mandelbrot discovers that all objects, even the simplest attributes of texture and shape, depend on our scale of observance. What shape is your car from five feet away? From five thousand feet away?
Mandelbrot creates the word “Fractal” and shows us an infinitely small line can fit in a finite space. Our lungs are like fractals. Lots of surface area packed into a small area.
“Scale is important.” Very important. Think about the universe all the way down to atoms. There is complexity and beauty at every scale.
Self-similarity is an old concept. It means at any scale, the part resembles the whole. Clouds are self-similar.
Simple equations can cause endless complex, beautiful patterns.
“Art that satisfies lacks scale, in the sense that it contains important elements at all sizes.”“
Strange Attractors
Traditional scientists ignored turbulence, the way fluids become erratic when they start to boil, for example. But what happens? Why does heated fluid roll to a point and then suddenly change into something unstable at the phase transition?
The boundary between smooth and turbulent was a mystery.
Swinney and Gollub tested fluid transitions in a laboratory by rotating fluids in a cylinder. When the fluid is rotated, donut-like patterns start to form. As they increased the speed of rotation, new frequencies appeared. At a certain point, this pattern breaks down and the fluid becomes chaotic.
Strange Attractors allow us to visualize random-seeming, complex behaviors in phase space, which can show beautiful patterns of loops and spirals..
Strange attractors have orderly orbits but never cross the same point again. If they did they would repeat the same cycle twice. How can an infinite pattern be drawn in a finite space? Look no further than the fractal for proof.
Attractors may show a couple of loops, for example. As you zoom in and look at the perimeter, you see multiple lines, each of which contains more lines and so on to infinity.
Cross sections of attractors show another beautiful pattern. We can slice it, sample one plane to see a new pattern, a beautiful crescent-shaped pattern. Henon calculated a similar crescent attractor using planetary motion as his model (google “Attactor of Henon” to see). He used a simple formula to create an infinitely complex, yet beautiful pattern. As you plot the Henon attractor it starts of random-seeming for a bit, then the pattern forms.
Universality
Feigenbaum combines math, physics and Romanticism. He looks for patterns in nature, in chaos.
Several brilliant chaos scientists looked to older, discarded science for fresh perspective. Maybe we got it wrong or ignored something worthwhile along the way.
Feedback loops can be used in equations. Feed the output of one calculation into the next and so on. We see feedback in electronic circuits and in nature.
“Concentrating on the boundary region between order and chaos… (Feigenbaum) knew this region was like the mysterious boundary between smooth flow and turbulence in a fluid.”
The transition from order to chaos exhibits bifurcations, the period-doublings that we see in animal populations. Feigenbaum sensed an order in this scaling and calculated the ratio of convergence, the constant for chaotic bifurcations at 4.66920.
Turbulence is “a continuous spectrum of frequencies.” Feigenbaum discovered the universality in disparate chaotic systems.
“You don’t really know that there isn’t another way of assembling all this information that doesn’t demand so radical a departure from the way in which you intuit things… There’s a fundamental presumption in physics that the way you understand the world is that you keep isolating its ingredients until you understand the stuff that you think is truly fundamental. Then you presume that the other things you don’t understand are details… One has to look for scaling structures - how do big details relate to little details… The only things that can ever be universal, in a sense, are scaling things… In a way, art is a theory about the way the world looks to human beings. It’s abundantly obvious that one doesn’t know the world around us in detail. What artists have accomplished is realizing there’s only a small amount of stuff that’s important, and then seeing what it was… I truly want to know how to describe clouds. But to say there’s a piece over here with that much density - to accumulate that much detailed information, I think is wrong. It’s certainly not how a human being perceives those things, and it’s not how an artist perceives things… Somehow the wondrous promise of the earth is that there are things beautiful in it, things wondrous and alluring, and by virtue of your trade, you want to understand them.” - Feigenbaum
The Experimenter
Libchaber was a low-temperature scientist, studying Helium at temperatures near absolute zero. He too read scientific works dating back to the 1600s, to get fresh ideas.